The number of subgroups of order 2 is odd.The number of normal subgroups of order 2 is odd.The number of p-core-automorphism-invariant subgroups (which in this case means the number of subgroups invariant under automorphisms in the 2-core of the automorphism group) of order 2 is odd.

In our case, the number of normal subgroups of order 2 is , which must be one of the numbers 1,3,7. For a non-abelian group, the socle cannot have order 16 or 32, so the number of normal subgroups of order 2 is exactly one.

Group

Second part of GAP ID

Hall-Senior number

Hall-Senior symbol

Nilpotency class

Number of subgroups of order 2

Number of normal subgroups of order 2

Number of 2-core-automorphism-invariant subgroups of order 2 (must be odd)

The number of subgroups of order 4 is odd.The number of normal subgroups of order 4 is odd.The number of p-core-automorphism-invariant subgroups (which in this case means the number of subgroups invariant under automorphisms in the 2-core of the automorphism group) of order 4 is odd.

index two implies normal (or more generally, any maximal subgroup of a group of prime power order is normal and has prime index)

The number of subgroups of order 4 equals the number of normal subgroups of order 4.

Group

Second part of GAP ID

Hall-Senior number

Hall-Senior symbol

Nilpotency class

Number of subgroups of order 4

Number of normal subgroups of order 4

Number of 2-core-automorphism-invariant subgroups of order 4 (must be odd)